As an extension of the MTP, Souza, Wilkens, and Martin [1]}
introduced the so called gauge invariant cumulants, which are
essential in the study of
polarization [2]}, [3]}, [4]}, [5]} as well
as charge transport, since they provide access to important related
quantities, such as the variance of the polarization [6]},
or the shift current [7]} (related to the third
cumulant). Patankar et al. [7]} use the ratio of the
third cumulant and the second as a gauge of nonlinearity in the
Su-Schrieffer-Heeger model [9]}. The Zak phase [10]} is
also the starting point to construct topological invariants, the
quantities which characterize the different phases of topological
insulators [11]}, [12]}, [13]}, [14]}, [15]}. The
MTP formalism has also been applied [16]} to study the
topological Haldane model.
